Endomorphism L-theory
Open books and automorphisms of manifolds were shown in Chaps. 28–30 to be closely related to the L-theory of the Laurent polynomial extensions A[z, z −1] of rings with involution A, with the involution extended by \(\bar z = {z^{ - 1}}\) . High-dimension
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Open books and automorphisms of manifolds were shown in Chaps. 28-30 to be closely related to the £-theory of the Laurent polynomial extensions A[z,z- 1] of rings with involution A, with the involution extended by:Z = z- 1. High-dimensional knots have been shown in Chaps. 31-33 to be closely related to the £-theory of the polynomial extensions A(s] of rings with involution A, with the involution extended by s = 1- s. This chapter deals with the £theory of A[x] with x = x, which is somewhat easier to deal with, yet shares many essential features with the £-theories of A[z, z- 1] and A(s].
f:
An endometric structure on a symmetric form (M, ¢)is an endomorphism M~M such that
) over A[x] with
Ps
=
(x- f)cp 8
+ 6f/Js-1
: C[x]n-r+s ---t C[x]r (r, s ~ 0) .
The skew-suspension (SC[x], P) is a f. g. projective (n + 2)-dimensional (-E)1 A[x]-Poincare complex over A[x]. The boundary symmetric
.n.:;
(D,lJ)
=
8(SC[x], !I>)
is a {homotopy) f.g. free (n +I)-dimensional (-E)-symmetric Poincare complex over A[x] corresponding to (C,f/J) and (f,6¢J) in (i), such that 'T[]+
(D)
r(D, 8)
= [C, !] E K1(A[x], n+) = Endo(A) = !I>( D) + (- )n!P(D)* E K 1 (A[x])
I
with respect to the canonical round finite structure on D. Proof (i) Let (C, f) be a finite chain complex of (A[x], [}+)-modules, i.e. a finite f.g. projective A-module chain complex C together a chain map f : C---tC. The f.g. projective A[x]-module chain complex D
=
e(x- j : C[x]---tC[x])
458
34. Endomorphism £-theory
is homology equivalent to (C, f), and every A-finitely dominated f.g. free A[x]-module chain complex D is chain equivalent to one of this type, with (D 1, ~) ~ (C, f). The Z[Z2]-module chain map
e(r;nd:
c ®A C--+C 181A C)
--+ D
i81A[xJ
D
defined by
e(rrd)r = (C ®A C)r EEl (C ®A C)r-1
=
----+ (D
®A[x]
D)r
EEl (C[x]
®A[x]
C[xDr-1 EEl (C[x]
(C[x]
®A[x]
C[x])r EEl (C[x)
®A[x]
®A[x]
C[x])r-1
C[x])r-2; (u,v)--+ (u,v,v,O)
is a homology equivalence inducing isomorphisms
so that there is defined an exact sequence rend
... --+ Qn+l(D, -t:)----+ Qn(C, t:) ~ Qn(C, -t:)--+ Qn(D, -t:)--+ ....
An (n + I)-dimensional ( -€)-symmetric structure () E Qn+l (D, -t:) on D is thus the same as ann-dimensional €-symmetric structure¢> E Qn(C, t:) with a refinement to an endometric structure (6¢>, ¢>) E Q~;':}(C, j, t:). Moreover
00
Hn+l-*(D)
:
~
) is a Poincare complex over A. 0 (ii)+(iii) Immediate from (i).
Definition 34.6 (i) Let U ~ End0 (A) be a *-invariant subgroup. The €symmetric endomorphism £-group LEnd[j(A, t:) (n 2: 0) is the cobordism group of finitely dominated n-dimensional t:-symmetric Poincare complexes (C, ¢>) over A with an endometric structure (J, 6¢>) such that [C, f] E U ~ Endo(A) . (ii) Let
U
=
Uo EEl V
~
Endo(A)
=
Ko(A) EEl Endo(A)
for some *-invariant subgroups U0 ~ K 0 (A), V ~ End 0 (A). The t:-symmetric -n reduced endomorphism £-group LEndv(A,t:) (n 2: 0) is -n
LEndv(A, t:) so that
=
ker(LEnd[j(A, t:)---+Lu0 (A, t:); (C, j, 6¢>, ¢>)--t(C, ¢>)) , -n
LEnd[j(A, t:) = Lifo (A, t:) EEl LEndv(A, t:) .
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